Haskell Design Patterns - Sample Chapter

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Design patterns and idioms can widen our perspective by showing us where to look, what to look at, and ultimately how to see what we are looking at. At their best, patterns are a shorthand method of communicating better ways to code (writing less, more maintainable, and more efficient code). This book starts with Haskell 98 and, through the lens of patterns and idioms, investigates the key advances and programming styles that together makeup "Modern Haskell". Your journey begins with the three pillars of Haskell. Then you'll experience the problem with Lazy I/O, together with a solution. You'll also trace the hierarchy formed by Functor, Applicative, Arrow, and Monad. Next, you'll explore how Fold and Map are generalized by Foldable and Traversable, which in turn are unified in a broader context by Functional Lenses. You'll delve more deeply into the Type system, which will prepare you for an overview of Generic Programming. Finally, you go to the edge of Haskell by investigating the Kind system and how this relates to Dependently-typed programming.

Who this book is written for

 Understand the relationship between the "Gang of Four" OOP Design Patterns and Haskell

 Explore the pervasive pattern of composition, from function composition through to high-level composition with Lenses  Synthesize Functor, Applicative, Arrow and Monad in a single conceptual framework  Follow the grand arc of Fold and Map on lists all the way to their culmination in Lenses and Generic Programming

P U B L I S H I N G

C o m m u n i t y

 Retrace the evolution, one key language extension at a time, of the Haskell Type and Kind systems

 Place the elements of Modern Haskell in a historical framework

D i s t i l l e d

Take your Haskell and functional programming skills to the next level by exploring new idioms and design patterns

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E x p e r i e n c e

Haskell Design Patterns

 Get a taste of Type-level programming in Haskell and how this relates to Dependently-typed programming

$ 34.99 US £ 22.99 UK

community experience distilled

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 Experience three ways of Streaming I/O: imperative, Lazy, and Iteratee-based

Ryan Lemmer

If you're a Haskell programmer with a firm grasp of the basics and ready to advance to modern idiomatic Haskell programming, then this book is for you.

What you will learn from this book

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Haskell Design Patterns

Haskell Design Patterns

Sa m

Ryan Lemmer

In this package, you will find:    

The author biography A preview chapter from the book, Chapter 1 'Functional Patterns – the Building Blocks' A synopsis of the book’s content More information on Haskell Design Patterns

About the Author Ryan Lemmer is software maker, coach, and strategic advisor based in Cape Town. With a background in mathematics and computer science and 20 years of developing software in the trenches, Ryan remains inspired and humbled by the process of creating and evolving software. Ryan is a polyglot programmer, who prefers to think in Haskell. He loves to learn and facilitate learning for others.

Preface This book is not a blow-by-blow translation of the Gang of Four design patterns (distilled out of the object-oriented programming paradigm). Having said that, wherever there is an intersection with Gang of Four patterns, we explore it more deeply. This book is also not intended as a definitive taxonomy of patterns in functional programming or Haskell. Instead, this book is the story of modern Haskell, one pattern at a time, one line of code at a time. By following the historical arc of development, we can place the elements of modern Haskell in a conceptual framework more easily.

What this book covers Chapter 1, Functional Patterns – the Building Blocks, explores the three pillars of Haskell, that is, first-class functions, lazy evaluation, and the Haskell type system, through the lens of patterns and idioms. We will cover some Gang of Four OOP design patterns along the way. Chapter 2, Patterns for I/O, explores three ways of streaming I/O, that is, imperative, lazy, and iteratee based. While you're at it, learn about the problem with lazy I/O, together with a solution. Chapter 3, Patterns for Composition, traces the hierarchy formed by functor, applicative, arrow, and monad, with a focus on how these types compose. Synthesize functor, applicative, arrow, and monad in a single conceptual framework. Chapter 4, Patterns of Folding and Traversing, demonstrates how fold and map are generalized by Foldable and Traversable, which in turn are unified in a broader context by functional lenses.

Preface

Chapter 5, Patterns of Type Abstraction, retraces the evolution of the Haskell type system, one key language extension at a time. We'll explore RankNtypes, existensial types, phantom types, GADTs, the type-case pattern, dynamic types, heterogeneous lists, multiparameter typeclasses, and functional dependencies. Chapter 6, Patterns of Generic Programming, delves into patterns of generic programming, with a focus on datatype generic programming. We will taste three flavors of generic programming: sum of products generic programming, origami programming, and scrap your boilerplate. Chapter 7, Patterns of Kind Abstraction, delves into the Haskell kind system and related language extensions: associated types, type families, kind polymorphism, and type promotion. We'll get a sense of type-level programming and then conclude by going to the edge of Haskell: diving into dependently-typed programming.

Functional Patterns – the Building Blocks Software design patterns were forged at a time when object oriented programming (OOP) reigned. This led to "design patterns" becoming somewhat synonymous with "OOP design patterns". But design patterns are solutions to problems, and "problems" are relative to the strengths and weaknesses of the context in which they occur. A design problem in OOP is not necessarily one in functional programming (FP), and vice versa. From a Haskell perspective, many (but not all) of the well known "Gang of Four" patterns [Design patterns, Gamma et al.] become so easy to solve that it is not worth going to the trouble of treating them as patterns. However, design patterns remain relevant for Haskell. "After al, as Erich Gamma said, "deja vu is language neutral" Modularity means more than modules. Our ability to de-compose a problem into parts depends directly on our ability to glue solutions together. To support modular programming, a language must provide good glue." - Why Functional Programming Matters - John Hughes

[1]

Functional Patterns – the Building Blocks

In order to have a meaningful conversation about Haskell design patterns, we'll begin our exploration by looking at the three primary kinds of "glue" in Haskell: first-class functions, the Haskell type system, and lazy evaluation. This chapter revisits the Haskell you already know through the lens of design patterns, and looks at: •

Higher-order functions

•

Currying

•

Recursion

•

Types, pattern matching, polymorphism

•

Lazy Evaluation

•

Monads

Higher-order functions Functions are our first kind of "glue" in Haskell.

Functions as first-class citizens Haskell functions are first-class citizens of the language. This means that: •

We can name a function just as we can name any primitive value: square = \x -> x * x

•

We can pass functions to other functions: map square [1, 3, 5, 7]

(Here, map is a higher-order function.) •

Functions can produce other functions (here, by currying the foldr function): sum = foldr (+) 0

•

Functions can form part of other data structures: let fs = [(* 2), (* 3), (* 5)] zipWith (\f v -> f v) fs [1, 3, 5]

This places Haskell functions on an equal footing with primitive types.

[2]

Chapter 1

Composing functions Let's compose these three functions, f, g, and h, in a few different ways: f, g, h :: String -> String

The most rudimentary way of combining them is through nesting: z x = f (g (h x))

Function composition gives us a more idiomatic way of combining functions: z' x = (f . g . h) x

Finally, we can abandon any reference to arguments: z'' = f . g . h

This leaves us with an expression consisting of only functions. This is the "point-free" form. Programming with functions in this style, free of arguments, is called tacit programming. It is hard to argue against the elegance of this style, but in practice, point-free style can be more fun to write than to read: it can become difficult to infer types (and, therefore, meaning). Use this style when ease of reading is not overly compromised.

Currying functions Haskell allows for both curried and uncurried functions: greetCurried :: String -> String -> String greetCurried title name = "Greetings " ++ title ++ " " ++ name greetUncurried :: (String, String) -> String greetUncurried (title, name) = "Greetings " ++ title ++ " " ++ name

Let's suppose that we need a function with the first argument fixed: greetCurried' :: String -> String greetCurried' = greetCurried "Ms" greetUncurried' :: String -> String greetUncurried' name = greetUncurried ("Ms", name)

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Functional Patterns – the Building Blocks

In both cases, we have applied one of the arguments and thereby specialized our original function. For the uncurried function we needed to mention all parameters in the reshaped function, while for the curried one we could just ignore subsequent arguments. Since it is fairly easy to translate a curried function to an uncurried function (and vice versa) a question arises: why and when would one want to use uncurried functions?

Currying and composability Consider a function that returns a few pieces of data, which you choose to express as a tuple: g n = (n^2, n^3)

Then suppose we want to find the maximum value in that tuple: max (g 11)

This would not work because max value is curried, but we can easily align the types by uncurrying: uncurry max (g 11)

Whenever we have a function returning a tuple and we want to consume that tuple from a curried function, we need to uncurry that function. Alternatively, if we are writing a function to consume an output tuple from another function, we might choose to write our function in uncurried (tuple arguments) form so that we don't have to later uncurry our function or unpack the tuple. It is idiomatic in Haskell to curry by default. There is a very important reason for this, as you will see with this example. Thanks to currying, we can do this: map (map square) [[1], [2,2], [3,3,3]]

We cannot, however, do this: let map' = uncurry map map' (map' square) [[1], [2,2], [3,3,3]]

We need to explicitly curry map' in order to compose it with other functions: (curry map') (curry map' square) [[1], [2,2], [3,3,3]]

Curried functions are composable, whereas uncurried functions are not.

[4]

Chapter 1

Decoupling with currying If we can apply one function argument at a time, nothing stops us from doing so at entirely different places in our codebase. For instance, we might "wire in" some authentication-related information into our function at one end of the codebase and use the specialized function in another part of the codebase that has no cognizance of the authentication argument and its related types. This can be a powerful tool for decoupling, the site of decoupling being the function argument list!

Recursion Recursion is even more fundamental than functions and types, in the sense that we can have recursive functions and types. Moreover, "recursion" can refer to syntax (a function or type referring to itself) or to the execution process.

Non-tail recursion Recursion can be viewed as a pattern for avoiding a mutable state: sumNonTail [] = 0 sumNonTail (x:xs) = x + (sumNonTail xs)

Without recursion, we would need to iterate through the list and keep adding to an intermediary sum until the list is exhausted, as shown in the following code: sumNonTail [2, 3, 5, 7]

This first expands into a nested chain of deferred operations, and when there are only primitives left in the expression, the computation starts folding back in on itself: ----------

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 17

sumNonTail [3, 5, 7] (3 + sumNonTail [5, 7]) (3 + (5 + sumNonTail [7])) (3 + (5 + (7 + sumNonTail []))) (3 + (5 + (7 + 0))) (3 + (5 + 7)) (3 + 12) 15

The sumNonTail function is non-tail-recursive. Because the recursion is "trapped" by the + operator, we need to hold the entire list in memory to perform the sum.

[5]

Functional Patterns – the Building Blocks

Tail recursion Tail recursion addresses the exorbitant use of space we have with non-tail-recursive processes: sumTail' acc [] = acc sumTail' acc (x:xs) = sumTail' (acc + x) xs sumTail xs = sumTail' 0 xs

This form of recursion looks less like mathematical induction than the sumNonTail function did, and it also requires a helper function sumTail' to get the same ease of use that we had with sumNonTail. The advantage is clear when we look at the use of "constant space" in this process: --------

sumTail [2, 3, 5, 7] sumTail' 0 [2, 3, 5, 7] sumTail' 2 [3, 5, 7] sumTail' 5 [5, 7] sumTail' 10 [7] sumTail' 17 [] 17

Even though sumTail is a recursive function, it expresses an iterative process. sumNonTail is a recursive function that expresses a recursive process.

Folding abstracts recursion Tail recursion is captured by the foldl function, as shown in the following code: foldlSum = foldl (+) 0

The foldl function expands in exactly the same way as sumTail'. In contrast, foldrSum expands in the same way as sumNonTail: foldrSum = foldr (+) 0

One can clearly see the tail recursion in the definition of foldl, whereas in the definition of foldr, recursion is "trapped" by f: foldr _ v [] = v foldr f v (x:xs) = f x (foldr f v xs) foldl _ v [] = v foldl f v (x:xs) = foldl f (f v x) xs

[6]

Chapter 1

Types, pattern matching, and polymorphism Algebraic types give us a very concise way to model composite types, even recursive ones. Pattern matching makes it easy to work with algebraic types. Type classes enable both the fundamental types of polymorphism: parametric and ad-hoc. Together, these capabilities allow us to easily express many of the Gang of Four patterns.

Algebraic types and pattern matching Algebraic data types can express a combination of types, for example: type Name = String type Age = Int data Person = P String Int -- combination

They can also express a composite of alternatives: data MaybeInt = NoInt | JustInt Int

Here, each alternative represents a valid constructor of the algebraic type: maybeInts = [JustInt 2, JustInt 3, JustInt 5, NoInt]

Type combination is also known as "product of types" and type alternation as "sum of types". In this way, we can create an "algebra of types", with sum and product as operators, hence the name Algebraic data types. By parameterizing algebraic types, we create generic types: data Maybe' a = Nothing' | Just' a

Algebraic data type constructors also serve as "deconstructors" in pattern matching: fMaybe f (Just' x) = Just' (f x) fMaybe f Nothing' = Nothing' fMaybes = map (fMaybe (* 2)) [Just' 2, Just' 3, Nothing']

On the left of the = sign we deconstruct; on the right, we construct. In this sense, pattern matching is the complement of algebraic data types: they are two sides of the same coin.

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Functional Patterns – the Building Blocks

Recursive types We capture the "composite pattern" very succinctly by creating recursive algebraic types, for example: data Tree a = Leaf a | Branch (Tree a) (Tree a)

This pattern describes the need to sometimes unify a composite structure with individual members of that structure. In this case, we're unifying Leaf (a leaf being a part of a tree) and Tree (the composite structure). Now we can write functions that act on trees and leaves: size :: Tree a -> Int size (Leaf x) = 1 size (Branch t u) = size t + size u + 1

Functions over recursive types are typically recursive themselves.

Polymorphism Polymorphism points at the phenomenon of something taking many forms. In Haskell, there are two kinds of polymorphism: parametric and ad-hoc (first described by Strachey in Fundamental Concepts in Programming Languages, 1967).

Parametric polymorphism In the following code, we have a function defined on a list of any type. The function is defined at such a high level of abstraction that the precise input type simply never comes into play, yet the result is of a particular type: length' :: [a] -> Int length' [] = 0 length' (x:xs) = 1 + length xs

The length object exhibits parametric polymorphism because it acts uniformly on a range of types that share a common structure, in this case, lists: length' [1,2,3,5,7] length' ['1','2','3','5','7']

In this sense, length is a generic function. Functions defined on parametric data types tend to be generic.

[8]

Chapter 1

Ad-hoc polymorphism "Wadler conceived of type classes in a conversation with Joe Fasel. Fasel had in mind a different idea, but it was he who had the key insight that overloading should be reflected in the type of the function. Wadler misunderstood what Fasel had in mind, and type classes were born!" - History of Haskell, Hudak et al. The canonical example of ad-hoc polymorphism (also known as "overloading") is that of the polymorphic + operator, defined for all Num variables: class Num a where (+) :: a -> a -> a instance Int Num where (+) :: Int → Int → Int x + y = intPlus x y instance Float Num where (+) :: Float → Float → Float x + y = floatPlus x y

In fact, the introduction of type classes into Haskell was driven by the need to solve the problem of overloading numerical operators and equality. When we call (+) on two numbers, the compiler will dispatch evaluation to the concrete implementation, based on the types of numbers being added: let x_int = 1 + 1 let x_float = 1.0 + 2.5 let x = 1 + 3.14

-- dispatch to 'intPlus' -- dispatch to 'floatPlus' –- dispatch to 'floatPlus'

In the last line, we are adding what looks like an int to a float variable. In many languages, we'd have to resort to explicit coercion (of int to float, say) to resolve this type of "mismatch". In Haskell, this is resolved by treating the value of 1 as a type-class polymorphic value: ghci> :t 1 -- Num a => a

1 is a generic value (a Num variable); whether 1 is an int variable or a float variable

(or a fractional, say) depends on the context in which it will appear.

[9]

Functional Patterns – the Building Blocks

Alternation-based ad-hoc polymorphism There are two kinds of ad-hoc polymorphism. We've seen the first type already in this chapter: data Maybe' a = Nothing' | Just' a fMaybe f (Just' x) = Just' (f x) fMaybe f Nothing' = Nothing'

The fMaybe function is polymorphically defined over the alternations of Maybe. In order to directly contrast the two kinds of polymorphism, let's carry this idea over into another example: data Shape = Circle Float | Rect Float Float area :: Shape -> Float area (Circle r) = pi * r^2 area (Rect length width) = length * width

The area function is dispatched over the alternations of the Shape type.

Class-based ad-hoc polymorphism We could also have achieved a polymorphic area function over shapes in this way: data Circle = Circle Float data Rect = Rect Float Float class Shape a where area :: a -> Float instance Shape Circle where area (Circle r) = pi * r^2 instance Shape Rect where area (Rect length' width') = length' * width'

Downloading the example code You can download the example code files from your account at http://www.packtpub.com for all the Packt Publishing books you have purchased. If you purchased this book elsewhere, you can visit http://www.packtpub.com/support and register to have the files e-mailed directly to you.

Instead of unifying shapes with an algebraic "sum of types", we created two distinct shape types and unified them through a Shape class. This time the area function exhibits class-based polymorphism. [ 10 ]

Chapter 1

Alternation-based versus class-based It is tempting to ask "which approach is best?" Instead, let's explore the important ways in which they differ: Alternation-based

Class-based

Different coupling between function and type

The function type refers to the algebraic type Shape and then defines special cases for each alternative.

The function type is only aware of the type it is acting on, not the Shape "super type".

Distribution of function definition

The overloaded functions are defined together in one place for all alternations.

Overloaded functions all appear in their respective class implementations. This means a function can be overloaded in very diverse parts of the codebase if need be.

Adding new types

Adding a new alternative to the algebraic type requires changing all existing functions acting directly on the algebraic "super type"

We can add a new type that implements the type class without changing any code in place (only adding). This is very important since it enables us to extend third-party code.

Adding new functions

A perimeter function acting on Shape won't be explicitly related to area in any way.

A perimeter function could be explicitly related to area by adding it to the Shape class. This is a powerful way of grouping functions together.

Type expressivity

This is useful for expressing simple type hierarchies.

We can have multiple, orthogonal hierarchies, each implementing the type class (For example, we can express multiple-inheritance type relations). This allows for modeling much richer data types.

Polymorphic dispatch and the visitor pattern While exploring ad-hoc polymorphism, we saw how we can simulate static type dispatch ("static" meaning that the dispatch is resolved at compile time, as opposed to "dynamic dispatch", resolved only at runtime). Let's return to our area function: area (Circle 10)

[ 11 ]

Functional Patterns – the Building Blocks

The preceding command will dispatch to the overloaded area function by matching: •

A sub type of the Shape algebraic type (subtype-based)

•

The type class to which Circle belongs that is, Shape (class-based)

We've referred to this as "dispatching on type" but, strictly speaking, type dispatch would have to resemble the following invalid Haskell: f v = case (type v) of Int -> "Int: " ++ (show v) Bool -> "Bool" ++ (show v)

Having said that, pattern-based and type-based dispatching are not that far apart: data TypeIntBool = Int' Int | Bool' Bool f :: TypeIntBool -> String f (Int' v) = "Int: " ++ (show v) f (Bool' v) = "Bool: " ++ (show v)

So far, we have only seen dispatching on one argument or "single dispatch". Let's explore what "double-dispatch" might look like: data CustomerEvent = InvoicePaid Float | InvoiceNonPayment data Customer = Individual Int | Organisation Int payment_handler :: CustomerEvent -> Customer -> String payment_handler (InvoicePaid amt) (Individual custId) = "SendReceipt for " ++ (show amt) payment_handler (InvoicePaid amount) (Organisation custId) = "SendReceipt for " ++ (show amt) payment_handler InvoiceNonPayment (Individual custId) = "CancelService for " ++ (show custId) payment_handler InvoiceNonPayment (Organisation custId) = "SendWarning for " ++ (show custId)

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Chapter 1

The payment_handler object defines behavior for all four permutations of CustomerEvent and Customer. In an OOP language, we would have to resort to the visitor pattern to achieve multiple dispatch. "Visitor lets you define a new operation without changing the classes of the elements on which it operates... Languages that support double or multiple dispatch lessen the need for the Visitor pattern." - Design Patterns, Gamma et al.

Unifying parametric and ad-hoc polymorphism On the one hand, we have parametric polymorphism, where a single generic function acts on a variety of types. This is in contrast to ad-hoc polymorphism, where we have an overloaded function that is resolved to a particular function by the compiler. Put another way, parametric polymorphism allows us to lift the level of abstraction, whereas ad-hoc polymorphism gives us a powerful tool for decoupling. In this sense, parametric polymorphism is considered to be "true polymorphism", while ad hoc is only "apparent" (or "synthetic"). Haskell blurs the distinction between ad hoc (specialized) and parametric (generic) polymorphism. We can see this clearly in the definition of the type class for equality: class Eq a where (==), (/=) :: a -> a -> Bool x == y = not (x /= y) x /= y = not (x == y)

(==) and (/=) are both given mutually recursive default implementations. An implementer of the Eq class would have to implement at least one of these functions; in other words, one function would be specialized (ad-hoc polymorphism), leaving the other defined at a generic level (parametric polymorphism). This is a remarkable unification of two very different concepts.

Functions, types, and patterns Functions and types intersect in several ways. Functions have a type, they can act on algebraic types, they can belong to type classes, and they can be specific or generic in type. With these capabilities, we can express several more Gang of Four patterns.

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Functional Patterns – the Building Blocks

The strategy pattern Thanks to higher-order functions (HOF), we can easily inject behavior: strategy fSetup fTeardown = do setup -– fullfil this function's purpose teardown

Here, we are defining an abstract algorithm by letting the caller pass in functions as arguments, functions that complete the detail of our algorithm. This corresponds to the strategy pattern, also concerned with decoupling an algorithm from the parts that may change. "Strategy lets the algorithm vary independently from clients that use it." - Design Patterns, Gamma et al.

The template pattern In "OOP speak", the strategy pattern uses delegation to vary an algorithm, while the template pattern uses inheritance to vary parts of an algorithm. In Haskell, we don't have OOP inheritance, but we have something far more powerful: type classes. We might easily abstract an algorithm with this type class that acts as an abstract class: class TemplateAlgorithm where setup :: IO a → a teardown :: IO a → a doWork :: a → a fulfillPurpose = do setup doWork teardown

"Define the skeleton of an algorithm in an operation, deferring some steps to subclasses. Template Method lets subclasses redefine certain steps of an algorithm without changing the algorithm's structure." - Design Patterns, Gamma et al.

[ 14 ]

Chapter 1

The iterator pattern "Provide a way to access the elements of an aggregate object sequentially without exposing its underlying representation" - Design Patterns, Gamma et al. The map function takes care of navigating the structure of the list, while the square function only deals with each element of the list: map square [2, 3, 5, 7]

We have decoupled flow control from function application, which is akin to the iterator pattern.

Decoupling behavior and modularizing code Whenever we pass one function into another, we are decoupling two parts of code. Besides allowing us to vary the different parts at different rates, we can also put the different parts in different modules, libraries, or whatever we like.

Lazy evaluation The history of Haskell is deeply entwined with the history of lazy evaluation. "Laziness was undoubtedly the single theme that united the various groups that contributed to Haskell's design... Once we were committed to a lazy language, a pure one was inescapable." - History of Haskell, Hudak et al Thanks to lazy evaluation, we can still consume the undoomed part of this list: doomedList = [2, 3, 5, 7, undefined] take 0 xs = [] take n (x:xs) = x : (take (n-1) xs) main = do print (take 4 doomedList)

The take object is lazy because the cons operator (:) is lazy, which is because all functions in Haskell are lazy by default. A lazy cons evaluates only its first argument, while the second argument, the tail, is only evaluated when it is selected. (For strict lists, both head and tail are evaluated at the point of construction of the list.) [ 15 ]

Functional Patterns – the Building Blocks

The proxy pattern has several motivations, one of which is to defer evaluation; this aspect of the proxy pattern is subsumed by lazy evaluation.

Streams The simple idea of laziness enables has the profound effect of enabling self-reference: infinite42s = 42 : infinite42s

Streams (lazy lists) simulate infinity through "the promise of potential infinity" [Why Functional Programming Matters, Hughes]: potentialBoom = (take 5 infinite42s)

A stream is always just one element cons'ed to a tail of whatever size. A function such as take consumes its input stream but is decoupled from the producer of the stream to such an extent that it doesn't matter whether the stream is finite or infinite (unbounded). Let's see this in action with a somewhat richer example: generate :: StdGen -> (Int, StdGen) generate g = random g :: (Int, StdGen) -- import System.Random main = do gen0 Int eval (Lit a) = a eval (Div a b) = eval a `div` eval b

The eval function interprets expressions written in our Expr data type: (eval (Lit 42)) –- 42 (eval (Div (Lit 44) (Lit 11)))

-- 4

Stripped of real-world concerns, this is very elegant. Now let's add (naive) capability to deal with errors in our interpreter. Instead of the eval function returning integers, we'll return a Try data type, which caters for success (Return) and failure (Error): data Try a = Err String | Return a

The refactored evalTry function is now much more syntactically noisy with case statements: evalTry :: Expr -> Try Int evalTry (Lit a) = Return a evalTry (Div a b) = case (evalTry a) of Err e -> Err e Return a' -> case (evalTry b) of Err e -> Err e Return b' -> divTry a' b' -- helper function [ 18 ]

Chapter 1 divTry :: Int -> Int -> Try Int divTry a b = if b == 0 then Err "Div by Zero" else Return (a `div` b)

The reason for the noise is that we have to explicitly propagate errors. If (evalTry a) fails, we return Err and bypass evaluation of the second argument. We've used the Try data type to make failure more explicit, but it has come at a cost. This is precisely where monads come into play. Let's make our Try data type a monad: instance Monad Try where return x = Return x fail msg = Err msg Err e >>= _ Return a >>= f

= Err e = f a

Next, we refactor evalTry by using the bind operator (>>=): evalTry' :: Expr -> Try Int evalTry' (Lit a) = Return a evalTry' (Div a b) = (evalTry' a) >>= \a' -> (evalTry' b) >>= \b' -> divTry a' b' -– evalTry' (Div (Lit 44) (Lit 0))

The bind operator enables error propagation: Err e

>>= _

= Err e

Whenever we have an Err function, the subsequent part of the bind chain will be ignored and will thereby propagate the error. While this gets rid of our case statements, it is hardly very friendly. Let's rewrite it using the do notation: evalTry'' (Lit a) = Return a evalTry'' (Div a b) = do a'